IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO.

2, APRIL 2009

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Optimal Cost-Beneﬁt for the Location of Capacitors in Radial Distribution Systems
H. M. Khodr, Member, IEEE, Zita A. Vale, Member, IEEE, and Carlos Ramos, Member, IEEE
Abstract—This paper proposes a computationally efﬁcient methodology for the optimal location and sizing of static and switched shunt capacitors in radial distribution systems. The problem is formulated as the maximization of the savings produced by the reduction in energy losses and the avoided costs due to investment deferral in the expansion of the network. The proposed method selects the nodes to be compensated, as well as the optimal capacitor ratings and their operational characteristics, i.e., ﬁxed or switched. After an appropriate linearization, the optimization problem was formulated as a mixed-integer linear problem, suitable for being solved by means of a widespread commercial package. Results of the proposed optimizing method are compared with another recent methodology reported in the literature using two test cases: a 15-bus and a 33-bus distribution network. For both cases tested, the proposed methodology delivers better solutions indicated by higher losses savings, which are achieved with lower amounts of capacitive compensation. To calculate exactly the energy savings and the deferral investment cost, a power ﬂow for radial distribution networks is executed before and after the compensation. The proposed method has also been applied for compensating an actual radial distribution network served by AES-Venezuela in the metropolitan area of Caracas. A convergence time of about 4 s after 22 298 iterations demonstrates the ability of the proposed methodology for efﬁciently handling compensation problems. Index Terms—Capacitor placement, cost-beneﬁt, distribution systems, mixed-integer linear programming.

Capacitor Investment Cost [$/kvar]. CRF Capital Recovery Factor. Loss Factor. Current magnitude that circulates through the line [A]. Active current component [A]. Reactive current component [A]. Investment in year 0 (Initial Investment). Cost coefﬁcient for losses and released capacity [$/kW/year]. Longitude between node and node . Project study period [year]. NPV PV Net Present Value of a cash ﬂow. Present Value of a stream of payments/savings. Electrical power losses due to Joule effect (kW). Reactive power [kvar]. Reactive power before compensation, obtained by means of load ﬂow [kvar]. Reactive power that circulates from node to node before compensation [kvar]. Reactive power installed in node because of the compensation [kvar]. Reactive power that circulates from node to node , after compensation [kvar]. Reactive power ﬂow through the lines by over dimensioning the capacitors banks [kvar]. Reactive power installed in node because of the compensation. This variable changes in function [kvar]. of the bank capacity Minimum reactive power obtained from the load curve [kvar]. Maximum reactive power obtained from the load curve [kvar]. Line resistance per km [ /km]. Line resistance .

NOMENCLATURE Present value of the cumulated beneﬁts over the project’s lifetime. Annual beneﬁts obtained in year . Loss Costs. Loss Costs before compensating. Loss Costs after compensating. Loss Costs in the branch between node and node . Power Capacity Charge [$/kVA/month]. Energy Charge [$/kWh].
Manuscript received February 15, 2008; revised June 23, 2008. Current version published March 25, 2009. This work was supported in part by FCT, FEDER, POCTI, POSI, POCI, POSC, and PTDC. Paper no. TPWRD-00123-2008. The authors are with the GECAD-Knowledge Engineering and DecisionSupport Research Center, Electrical Engineering Institute of Porto-Polytechnic Institute of Porto (ISEP/IPP), Porto 4200-072, Portugal (e-mail: hmk@isep. ipp.pt; zav@isep.ipp.pt; csr@isep.ipp.pt). Digital Object Identiﬁer 10.1109/TPWRD.2008.2005391

and simulated annealing [27] have been proposed for optimally placing capacitive compensation in distribution networks. discrete capacitor sizes. and the constraints of selecting for each node only one among the various proposed capacitors banks sizes and types (ﬁxed or switched).g. optimization methodologies based on artiﬁcial intelligence.).
I. The proposed algorithm allows the evaluation of a higher number of constraints inherent to the large-scale optimization problems faced by actual utilities. [13]. voltage limits. the reactive power balance in each node of the network. 24. switched capacitors and switching times. NO. The second group represents the decision of installing or not a bank of capacitors in a speciﬁc point. For a comprehensive review of the abundant literature on the subject the reader is further referred to the literature surveys conducted in [29]–[31]. subject to the whole constraint set of the optimal reactive power ﬂow. network operational constraints (e. as well as the economical value of released feeder capacity due to compensation. e. genetic algorithms [18]. Basically.. A myriad of algorithms of very different nature and degree of sophistication addressing this topic are available in the published literature. artiﬁcial neural networks [22]. In order to cope with the problem size and overcome the inherent complexity of programming methods. network operational constraints (e.e. Due to computing power increase. However. Two types of variables take part in the formulation: the ﬁrst group of variables represents physical magnitudes such as power ﬂow through the network. The delivery of power from sources to the consumer points is always accompanied of power losses. switched capacitors and switching times. 2. operation and maintenance costs of the installed compensation.788
IEEE TRANSACTIONS ON POWER DELIVERY. evolutionary programming [19]. Number of capacitor bank size considered. As a consequence of the deregulation process. [14] have been applied to solve the optimal capacitor allocation and sizing problem. Nevertheless. voltage. The optimization problem has been formulated as the maximization of the total savings produced by the reduction in energy losses and the avoided costs due to investment deferral in the expansion of the network over a considered period. Most of them analyze the localization and sizing of static and switched capacitors in radial distribution systems. few of these methodologies take into account the total costs linked to reactive power compensation.1) to install a bank node . such as a uniform load distribution along the feeder. Early efforts relied on analytical approaches that delivered simple closed-form solutions to the optimization problem [7]–[10]. radial feeder with laterals. In general. Impedance angle. they are real variables. quadratic programming [12]. load ﬂow limits. Thus. operation and maintenance costs.
Dynamic programming [11]. [23] fuzzy sets and fuzzy dynamic programming [24]. The optimal reconﬁguration model responds to changes in the network topology by switching the automatic breakers installed in the grid. besides including the basic constraints of the electricity laws. nonlinear capacitor costs. 2009 at 09:53 from IEEE Xplore. [25].. some heuristic methods have been devised [15]–[17]. discrete capacitor sizes. uniform feeder size and radial feeder with no laterals.g. Restrictions apply. VOL. the proposed analytical methods required seemingly unrealistic assumptions. More recently. as well as the costs of energy losses and released capacity due to decreasing peak power losses associated to the reactive component of branch currents. The problem of optimal capacitor allocation and sizing for maximizing cost savings has called the attention of researchers for several decades. radial feeder with laterals. the distribution business has remained as a regulated monopoly. installation.). Downloaded on July 29. Some other proposals combine heuristic/metaheuristic techniques with programming methods [28]. capacity Discount rate [%/year]. Such nonnegligible amount of losses has a direct impact on the ﬁnancial results and the overall efﬁciency of distribution utilities.. distribution utilities are currently facing increased pressures from shareholders and regulatory authorities to improve investment and operational efﬁciency. The methodology was developed only for radial networks because most of systems are deployed in a radial basis due to their simple operation and low investment costs. The costs formulation includes investment. the optimization problem was formulated in a more sophisticated manner and solved by numerical programming techniques. Some studies have determined that power losses occurred in distribution networks due to Joule effect can account for as much as 13% of the generated energy. the model is ﬂexible enough for including
Authorized licensed use limited to: MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY. Number of nodes in the network. Losses can be further reduced by connecting capacitors in series or parallel (shunt) to locally supply a considerable portion of the reactive power demanded by the consumers and thereby reducing the reactive component of branch currents [4]–[6].
. [20] tabu search [21]. This paper proposes a computationally very efﬁcient methodology for an optimal location and sizing of static and switched shunt capacitors in radial distribution networks. Therefore. etc. APRIL 2009
Binary decision variable (0. etc. which under some premises are linearized and solved through mixed-integer linear programming. load ﬂow limits. i. The installation of shunt capacitors provides supplementary beneﬁts. such as improvement of the voltage proﬁle. they are binary variables. and nonlinear mixed-integer programming [4]. the power factor and the stability of the distribution system [1]. The proposed model has been reﬁned in order to consider varying loads. The mathematical formulation of the optimization problem starts with the electrical variables of the circuit under study and its topology. INTRODUCTION
A
FTER deregulation of the power industry. Number of lines in the network. active losses in distribution systems can be reduced by optimal reconﬁgurations of the network [1]–[3]. sophisticated algorithms consider in their formulations one or more of the following modeling reﬁnements: varying loads.g. expert systems [26].. methods for loss reductions that optimally allocate scarce ﬁnancial resources and maximize ﬁrm value are essential for achieving the ﬁnancial goals of distribution companies.

II.KHODR et al. the global cost equation can be obtained (13)
Authorized licensed use limited to: MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY. the power losses linearized as a function of the reactive power or apparent power are expressed by the following functions: (9) (10) (11) The expected savings by placing capacitor banks are modthat includes eled by means of an annual cost coefﬁcient. In Section V. power losses can be rewritten as follows [1]: (3) The active current component depends only on the circuit load.0 p. regulations requirements and subject to penalizations. and hence. Theoretical Basis for the Model Formulation The following expression determines the electric power losses because of the Joule effect in a line: (1) The magnitude of the circulating current is given as (2) By replacing (2) in (1). operational restrictions preventing the installation of capacitors in some nodes of the network. by multiplying (12) and (9). This requirement can be fulﬁlled according to the following procedure. In order to apply mixed-integer linear programming. economic assessments of the solutions provided by the algorithm are analyzed.: OPTIMAL COST-BENEFIT FOR THE LOCATION OF CAPACITORS IN RADIAL DISTRIBUTION SYSTEMS
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other constraints. it could be modiﬁed by injecting capacitive reactive power through the installation of capacitor banks. Equation (4) can be decomposed as follows: (5) where (6) One of the components lowing equation: can be represented by the fol-
(7) . B. • The nodes allowed to be compensated are known according to the all constrains as well as the type of bank capacity that would be installed. Model Assumptions The optimal location and sizing of shunt capacitors in the electric distribution network is a nonlinear problem. only the reactive component that generates technical losses [1] will be further considered (4)
depends on the inductive reactive The reactive current power demanded by the system. The article is concluded in Section VI. the main assumptions to be considered are as follows. a suitable linearization of the technical losses in (4) is necessary. • Three-phase balanced systems. Restrictions apply. After substituting the expressions (7) and (8) in (5). the optimization problem and constraints are mathematically formulated. as it will be seen next. The in $/kWh. such as proprietary company rules or standards. However. It must be delivered to the customer to cover their consumption. cost of losses is related to the energy cost Finally. CAPACITOR PLACEMENT A. 2009 at 09:53 from IEEE Xplore. and it could be modiﬁed by other methodologies distinct from compensation. Section IV presents numerical results of the comparison with other recently proposed methodology as well as for a compensation problem. can be obtained by running an optimal power ﬂow. This value can be modiﬁed by placing banks of capacitors. Downloaded on July 29. The other component of the current is replaced by the following expression: (8) where stands for the actual reactive power ﬂow circulating in the branches of the network. This paper is organized as follows: Section II provides the theoretical basis and main assumptions for building a model of optimal capacitor placement. In Section III.. two parts: costs associated with annual savings due to released capacity and costs associated with annual energy losses (12) Released capacity factor is assessed though a capacity charge in $/kVA/month related to the ﬁxed cost of the network. • The phase shift between the system bus voltage angles is close to zero. this problem can be (under certain approximations) linearized and solved by using mixed-integer linear programming.
. or a simply approximate power ﬂow for radial distribution systems in the actual state of the distribution circuit. etc. The purpose is to locate these capacitors in the points where they can improve best the technical and economic circuit performance. • The bus voltage magnitudes are set equal to 1. Before starting with the development of the mathematical framework. which are beyond the scope of this work. the initial reactive power ﬂow. Therefore. in the ﬁrst iteration of the power ﬂow.u.

Reactive power injections at node j . NPV of the compensation investment project.790
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(22) subject to the following constraints. costs before compensating that initially ﬂows through the netThe reactive power work. The compensation project will be proﬁtable when the Present Value (PV) of the beneﬁts associated with the loss reduction. i. NO. which is equivalent to the maximization of the project prof. which cannot be modiﬁed. 1.
. The objective function will ﬁnally be rewritten as follows:
(18) The ﬁrst term of (18) corresponds to the initial power losses. the savings are the difference between the and the costs after compensating. 2). 2009 at 09:53 from IEEE Xplore. 2.
In this way. the initial objective function to be minimized will be stated as follows:
III. it can be rewritten as
C. The annual project beneﬁts are obtained itability from the savings produced by the reduction in energy losses and the avoided costs due to investment deferral in the expansion of the network that can be veriﬁed as a reduction in the maximum power metered during the entire year. the Net Present Value (NPV) is positive (see Fig.e. transporting and installing all capacitor banks selected by the algorithm according to their capacity . the project lifetime and the discount factor (which should adequately reﬂect the ﬁrm opportunity cost of capital) have to be deﬁned according to the company’s ﬁnancial policy. 1). and [33]. will be affected when compensation in bus ..e. Under these conditions.
Authorized licensed use limited to: MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY. VOL. PROBLEM FORMULATION The objective is to minimize the total cost of the project. [32]. Restrictions apply. For the economic assessment of the investment project. multiplied by a binary deciassociated with the placing decision sion variable (21) stands for the total investment cost of the capacitors per unit of kvar. Investment Appraisal The economic evaluation of the project must take into account the necessary investments for the installation of capacitors and the beneﬁts obtained in each period from lower losses. Downloaded on July 29. 2.
Fig. as follows: (14) (19) where (15) The equation of the project proﬁtability (14) can be rewritten as follows: (16) (20) The investment cost is the aggregate cost of purchasing. APRIL 2009
Fig. 24. calculated by means of the methodologies such as those proposed in [2]. over the considered project lifetime is greater than the initial investment in the capacitors banks.. In other words. Therefore. is included. we could say that the beneﬁts to be obtained annually might be computed by the following expression: (17) By replacing (17) in (16). The present value of losses savings can be computed by means of the capital recovery factor (CRF) assuming uniform beneﬁts (annuities) along the periods. the inequality holds (see Fig. i.

The optimizing algorithm attains a losses reduction of 28. reactive power balance conditions must be fulﬁlled according to the following expression (see Fig.7 kW is obtained with the installation of 1193 kvar in nodes 3 and 6. respectively. If the capacitor bank to be installed is ﬁxed: (26) If the capacitor bank to be installed is switched
TABLE I 15-BUS TEST NETWORK: INITIAL CONDITIONS AND RESULTS OF [1]
TABLE II 15-BUS TEST NETWORK: FINAL SIMULATION CONDITIONS OF THE METHODOLOGY
TABLE III 15-BUS TEST NETWORK: FINAL CONDITIONS OF THE METHODOLOGY
system in the metropolitan area of Caracas served by AES Corporation in Venezuela. The data of the system are obtained from [34]. 4. The conditions of the networks after installing the capacitors are presented in Table III. 2009 at 09:53 from IEEE Xplore. from the load curve of the circuit. Single-line diagram of the 15-bus distribution network. The initial condition of the network (before compensating) is shown in Table I. Reactive power balance at node j . In the simulation done in [1]. the following expression needs to be satisﬁed: (24) Constraint 3: Power Control in Lines: In order to avoid any inverse reactive power ﬂow through the lines by over dimensioning the capacitors banks.58 kW with the installation of 900 kvar in nodes 3.KHODR et al. 3. 15-Bus Test Network
(27) and are minimal and maximal reactive power where obtained. the algorithm has been implemented for the compensation problem of a radial distribution
The single-line diagram of the 11 kV. hence. the condition of installing only one capacitor bank per node is imposed. 4.: OPTIMAL COST-BENEFIT FOR THE LOCATION OF CAPACITORS IN RADIAL DISTRIBUTION SYSTEMS
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Fig. The nodes where the capacitors were ﬁnally located and their compensating capacities are shown in Table IV. iterations and convergence time are presented in Table II. 6.
Authorized licensed use limited to: MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY. Restrictions apply. System data are obtained from the same source described in [1]. and 11. The comparison between the proposed methodology and the results provided by [1] are shown below. IV. 3):
Fig. In addition.
Constraint 1: Reactive Power Balance (Kirchhoff’s Law): For each node . the following constraint should be added: (25) Constraint 4: Determination of the Type of Capacitors (Fixed or Switched): From the representative load curve the minimum and maximum reactive power can be computed with the aim of determining if the capacitors being installed at a speciﬁc bus are ﬁxed or controlled according to the following conditions.
(23) Constraint 2: Selection of a Unique Bank Per Node: On the other hand. 4.
. Downloaded on July 29. A. a reduction of 27. The results are compared with the solutions obtained with the methodology presented in [1]. 15-bus distribution system is shown in Fig. RESULTS The proposed methodology has been initially applied to a 15-bus and 33-bus distribution test networks. The number of variables.

. and 600 kvar. 5. 2009 at 09:53 from IEEE Xplore. 13.66 kV. Downloaded on July 29. The capacities and sizes of the capacitors that were considered in the optimization problem are the existing in stock in the warehouse of the AES Electric Company.
This network includes all equipment of sectioning. Table XVI presents the network data. 8. Restrictions apply. The large measurement campaign was promoted by the AES Electrical Company. The ﬁnal conditions of the network are shown in Table VII. The selected nodes where the capacitors were ﬁnally located and their respective capacities are shown in Table VIII. By applying the proposed algorithm a losses reduction of 88. a reduction of 79. Fig. NO. their connectivity and the numbering of the nodes. In order to accelerate the calculations. 24. Table IX provides the numerical values of the parameters for which the network has been simulated. 7. 6 depicts the representative daily load pattern served by this system. and 30. 300. 33-bus system is illustrated in Fig. 2. transformation and switching of the lateral. Single-line diagram of the 33-bus test system. These capacitors sizes are of 150.91 kW is achieved with the installation of 1350 kvar in nodes 6. 29. iterations and convergence time are presented in Table VI. C. 23.792
IEEE TRANSACTIONS ON POWER DELIVERY. 33-Bus Test Network The single-line diagram of the 12. 5. The number of variables. The initial conditions of the network before compensating are presented in Table X. The single-line diagram of the network is provided in Fig. 27. The comparison between the proposed methodology and the results obtained in [1] are shown below. The initial conditions of the network are shown in Table V. The most representative measurements of the distribution circuit and each network node have been requested to the electrical utility to carry out the optimal capacitors location study into distribution network. The load proﬁles were obtained by a number of measurements and statistical evaluation
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TABLE IV 15-BUS TEST NETWORK: LOCATION AND CAPACITY
TABLE VII 33-BUS TEST NETWORK: FINAL CONDITIONS OF THE METHODOLOGY
TABLE VIII 33-BUS TEST NETWORK: LOCATION AND CAPACITY
Fig. this network has been reduced to 141 nodes by eliminating the sectioning devices. The data of the system are obtained from [2].
TABLE IX 141-BUS TEST NETWORK: SYSTEM DATA
TABLE V 33-BUS TEST NETWORK: INITIAL CONDITIONS AND RESULTS OF [1]
TABLE VI 33-BUS TEST NETWORK: FINAL SIMULATION CONDITIONS OF THE METHODOLOGY
TABLE X 141-BUS NETWORK: STATE OF THE CIRCUIT BEFORE REACTIVE COMPENSATION
B. VOL. In the simulations done in [1].5 kW is obtained with the installation of 1400 kvar in nodes 12 and 29. Real Test Case The developed method was applied to a real network of 231 nodes that covers a zone of the metropolitan area of Caracas.

The results on the 15-bus and 33-bus test cases were compared with the results computed with a recent methodology reported in [1]. ECONOMIC EVALUATION The proposed method was tested on three distribution systems consisting of 15. thus contributing to ensure the robustness of the whole methodology. it is important to stress that the results do not depend on the modeling language (CPLEX) and in fact they would be the same using other optimization platform. Downloaded on July 29. The methodology has also applied to a real case of 141 buses corresponding to a circuit that feeds a zone of the metropolitan area of Caracas
Authorized licensed use limited to: MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY. The load growth could be considered by means of an adequate demand forecast method if the network was not saturated. These models were solved using the commercial package CPLEX [35]. The nodes where capacitors were ﬁnally located and their respective capacities are included in Table XIII. Load curve of the circuit. 2009 at 09:53 from IEEE Xplore. All the cases are executed in a computer equipped with Windows XP Operation System with an Intel Core 2 Duo T7200 Processor and 2 GB of RAM. CAPACITY AND TYPE OF INSTALLED CAPACITORS
TABLE XIV COMPARISON OF THE ECONOMIC PERFORMANCE FOR THE 33-BUS TEST CASE
Fig. It was found that the results of this methodology are considerably better than the results provided in the aforementioned references. In any case. where the nodes are saturated respect to load growth.: OPTIMAL COST-BENEFIT FOR THE LOCATION OF CAPACITORS IN RADIAL DISTRIBUTION SYSTEMS
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TABLE XIII 141-BUS NETWORK: LOCATION. In any case.
TABLE XI 141-BUS TEST NETWORK: FINAL CONDITION OF THE SIMULATION
TABLE XV ECONOMIC EVALUATION OF THE 141-BUS TEST CASE
TABLE XII 141-BUS TEST NETWORK: STATE OF THE CIRCUIT AFTER REACTIVE COMPENSATION
to determine a daily load curve for each node and each distribution feeder circuit leaving the substation. A losses reduction of 70. 6. Table XII presents the ﬁnal condition of the network after compensating with the developed optimization model. and 141 buses. re-
spectively. Restrictions apply. reducing the energy losses and optimizing the investments in reactive power compensation. V. and 94. 55. A new study could be carried out every two or three years to locate new capacitors in the network. 79. 50. The number of intervening variables. because these days introduce noise in the daily load pattern.74 kW is accomplished with the installation of 1500 kvar ﬁxed and 300 controlled distributed in the nodes 23. it should be referred that the developed formulation can be easily adapted to other commercial packages available in the market. is widely known and well established.KHODR et al. All days of the week were considered excepted the Saturday and Sunday days. It is also important to refer that the adopted optimization technique.
. iterations and convergence time is included in Table XI. 64. CPLEX [35]. 33. The study was carried out in the distribution circuit belonging of the metropolitan area of Caracas.

In both cases. NO. 24. Single-line diagram of a real distribution network in the metropolitan area of Caracas. 2. TABLE XVI NETWORK AND CONNECTIVITY DATA
attended by AES-Venezuela.794
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. the results are compared with the results of [1]. 2009 at 09:53 from IEEE Xplore. APRIL 2009
Fig. VOL. A short convergence time of about 4 s after 22 988 iterations shows the effectiveness of the proposed mathematical model for solving the optimal compensation problem in a framework of high dimensionality. Downloaded on July 29. In the 33-bus case. 7. the execution of a load ﬂow
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An economic evaluation of results is performed for two of the cases presented previously: the 33-bus and the 141-bus distribution systems. Restrictions apply.

His current research activities are planning. D. M. 1. 17. vol.. Restrictions apply. and distributed generation. Ng. H. 1. He was an Associate Professor of electrical engineering at Universidad Simón Bolívar. Eng. [34] D.” Proc. P. 33–38. M. respectively. APRIL 2009
[31] H. Vale (M’93) received the diploma in electrical engineering and the Ph. and A. Power Del. vol. Portugal.. “Efﬁcient load ﬂow method for distribution systems with radial or mesh conﬁguration. Das. France.D. 1995.D. Elect. Jan. He has participated in a number of projects performed for the local industries. S. no. Elect. pp. and M. He is a Coordinator Professor of Computer Engineering at the Polytechnic Institute of Porto/Institute of Engineering. power quality. Portugal. 335–346. “Simple and efﬁcient method for load ﬂow solution of radial distribution networks. 1994.” Proc.D. respectively. Kothari. Jul. no. pp. Her main research interests concern artiﬁcial intelligence (AI) applications to power system operation and control. vol. Porto. She is involved in several R&D projects concerning the application of AI and decision-support techniques to engineering problems. Currently. P. 2009 at 09:53 from IEEE Xplore. in 1986 and 1993. and A. 24. degree in electrical engineering from the University of Porto. Inst.Sc.
Carlos Ramos (M’93) received the diploma and the Ph. She is a Coordinator Professor of Power Systems at the Engineering Institute—Polytechnic Institute of Porto (ISEP/IPP). Haque. [35] ILOG AMPL CPLEX System User’s Guide. 15. Power Energy Syst. 2. Eng.
. Chikhani. A.
Authorized licensed use limited to: MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY. respectively. 5. 143.” Elect. operation. [32] D. VOL. pp. 4. Das./Eng.. Kothari. Portugal. pp. Y. His main R&D interests are artiﬁcial intelligence and decision support systems. H. Khodr (M’99) received the Ph. grounding systems.” IEEE Trans. degree in electrical engineering from the University of Porto. electricity markets. Kalam. H. he is a Researcher at GECAD—Knowledge Engineering and Decision-Support Research Group of the Electrical Engineering Institute of Porto—Polytechnic Institute of Porto (ISEP/IPP). Salama. Inst. 1996. 8. 141..0 Ed. vol. Cuba.
Zita A. degrees in electrical engineering from the José Antonio Echeverría Higher Polytechnic Institute (ISPJAE). 387–392. Jan.796
IEEE TRANSACTIONS ON POWER DELIVERY. He was a Researcher at INESC Porto. and optimization. no. Downloaded on July 29. in 1986 and 1993. Nagi. “Classiﬁcation of capacitor allocation techniques. 2002. Porto. “Novel method for solving radial distribution networks. 2000. no. and D. [33] M. and economics of electrical distribution and industrial power systems. Havana. N. NO.. ILOG. M. in 1997 and 1993. Venezuela. 291–298.